The tenants will be given an application form for Rent Supplement, and part of that form must be completed by the landlord (either one of your parents can sign). The landlord must also supply their PPSN or Tax Reference Number as part of the application. The form will also ask if there are multiple names on the lease or tenancy agreement, and as oldnick said, it is best to give accurate information regardless of what the tenant has requested.
There is a maximum rent limit in place for persons applying for Rent Supplement - if the rent exceeds the amount for the county and family size, rent supplement is unlikely to be granted. Your parents might be asked by the tenants to reduce the rent if it exceeds the limit.
Are the couple renting an entire house (3 or 4 bed?) - they may be considered over-accommodated, which might also lead to refusal of Rent Supplement.
Depending on the area, there can be a delay in processing Rent Supplement applications. Your parents should be very careful in letting the rent "slide" during this time - especially if there's a possibility of the supplement being refused.